Question

Belle is making necklaces. She has 42 yellow beads and 36 brown beads. If Belle wants to make all the necklaces exactly the same with no beads left over, what is the greatest number of necklaces she can make?

Answer 1

To find the greatest number of necklaces Belle can make, we need to determine the greatest common divisor (GCD) of the number of yellow beads (42) and the number of brown beads (36).

We can use the prime factorization method to find the GCD:

1. **Factor 42:**
\[
42 = 2 \times 3 \times 7
\]

2. **Factor 36:**
\[
36 = 2^2 \times 3^2
\]

3. **Identify the common prime factors:**
- The common prime factors between 42 and 36 are \(2\) and \(3\).

4. **Determine the minimum exponent for each common prime factor:**
- For \(2\), the minimum exponent in 42 is \(1\) (since \(2^1\) is present) and in 36 is \(2\) (since \(2^2\) is present). The minimum exponent is \(1\).
- For \(3\), the minimum exponent in 42 is \(1\) (since \(3^1\) is present) and in 36 is \(2\) (since \(3^2\) is present). The minimum exponent is \(1\).

5. **Calculate the GCD:**
\[
\text{GCD} = 2^1 \times 3^1 = 2 \times 3 = 6
\]

Thus, the greatest number of necklaces Belle can make, with no beads left over, is \( \boxed{6} \).

To confirm:
- If she makes \(6\) necklaces, she will use:
- Yellow beads: \( \frac{42}{6} = 7\) yellow beads per necklace
- Brown beads: \( \frac{36}{6} = 6\) brown beads per necklace

This distribution utilizes all beads and confirms that the solution is correct.